Dynamic Allocation of Subcarriers and Powers in. a Multiuser OFDM Cellular Network

Transcription

1 Dynamic Allocation of Subcarriers and Powers in 1 a Multiuser OFDM Cellular Network Thaya Thanabalasingham, Stephen V. Hanly and Lachlan L. H. Andrew Abstract This paper considers a resource allocation problem in the downlink of a multiuser, multicell OFDM cellular network with frequency selective fading and imperfect channel state information. Each user has a given data rate requirement. The task is to determine power and subcarrier allocations for users to ensure that users outage probabilities are low. A two-layer approach is used. The higher layer allocates powers based on the average gains of the links, with cells coupled via the mutual interference between them. To improve the performance in frequency selective fading, the lower layer allocates discrete subcarriers to the users in a way which takes into account statistical knowledge about the subcarriers. Index Terms Cellular network, orthogonal frequency division multiplexing (OFDM), power control, resource allocation I. INTRODUCTION Orthogonal Frequency Division Multiplexing (OFDM) is an important technique for communicating over frequency selective channels. By dividing the available bandwidth into orthogonal, noninterfering subcarriers and adopting a parallel transmission strategy, it offers better immunity to the multipath fading effects of the wireless channel than single carrier transmission systems. OFDM is widely deployed in commercial systems such as xdsl modems [1], [2] and low mobility wireless LANs [3]. It is also a strong candidate for the future wireless cellular systems. T. Thanabalasingham and S. V. Hanly are with ARC Special Centre for Ultra Broadband Information Networks (CUBIN), Department of Electrical and Electronic Engineering, University of Melbourne, Australia. L. L. H. Andrew is with Department of Computer Science, California Institute of Technology, California, USA. This work was supported by the Australian Research Council (ARC).

2 2 Although OFDM typically multiplexes low rate data substreams from a single user onto all the subcarriers, in the downlink of an OFDM cellular network it can also be used to multiplex the data streams from different users onto subsets of the subcarriers. This paper considers the resource allocation problem in such a multiple user, multiple cell OFDM system, and we focus on the downlink. Users in the same cell do not interfere with each other, but there is cross-talk between users in different cells. We consider the joint problem of allocating subcarriers and powers to users within each cell, subject to meeting data rate requirements of the users. We assume that there are several users in each cell. This models a wideband cellular OFDM system with full or aggressive frequency reuse so that co-channel interference is significant. This work assumes that the base stations have statistical knowledge of the channel conditions, but not the instantaneous channel values. In this setting, no resource allocation can guarantee that users receive the target data rates; instead, we focus on outage probability, the fraction of time for which the rate is not achieved. Our resource allocation problem is to determine power and subcarrier allocations to minimize the variation in the outage probabilities experienced by different users. A central idea of this paper is that if transmit powers are set a priori, then further optimization of subcarrier allocations for a given user can be performed without the need to keep track of interference from other users. The other-cell users can update their subcarrier allocations as they wish, without affecting the given user. To make this method work, we propose a two-stage approach. In the first stage, base station transmit powers are calculated, considering the impact of interference on all users. The aim in the first stage is to select transmit powers in a way to minimize the total transmit power used in the network. In the second stage, subcarriers are allocated to the links taking into account the statistics of the fast fading and the amount of averaging each user can perform. The objective for this layer is to minimize the maximum outage probability within the cell. A special feature of the scheme that we derive in this paper is that the transmit power spectral density (PSD) of each base station is flat across the frequency band. Only the total power level is varied

3 3 across base stations. This constraint allows us to use the lower-layer subcarrier allocation algorithm proposed in this paper. However, one might be concerned that it also constrains performance. For this reason, we compare the performance of our scheme with two other schemes. One of these schemes ( Dynamic ) removes this constraint and hence frees up the degrees of freedom in the higher-layer optimization. The other scheme we compare with ( Static ) allocates subcarriers in proportion to user data rate targets, and then does power control given these fixed bandwidth allocations. II. RELATED WORK Resource allocation in OFDM systems has received considerable attention in the literature. The optimal single user resource allocation is obtained by water pouring [4]. It is also possible to perform multiuser resource allocation by using iterative water pouring techniques [5]. However, the iterative water-pouring approach does not, in general, guarantee fairness among the users. The resource allocation problem becomes combinatorial in nature when the fairness criterion is included into the problem, leading to a search for suboptimal solutions. This holds true even in the case of single cell [6], [7]. In the context of a multi-cell network, the resource allocation problem is further complicated by inter-cell interference. This has been formulated [8] with the objective of minimizing power levels, subject to rate constraints on the individual links. The subcarriers were allocated to users in a heuristic fashion. Iterative power control was performed on each assigned subcarrier, in order to select the transmit powers for users assigned to that subcarrier. In the bandwidthconstrained power minimization problem [9], an upper bound is imposed on the number of subcarriers to each user as a way of minimizing the mutual interference. All of the work above assumes perfect channel state information (CSI) at the transmitters, which may not be realistic, especially when the channel conditions vary quickly with time. In this case, the resource allocation needs to be performed based on the statistical knowledge of the channel conditions. One such resource allocation problem has been studied [10] in a single user context with the objective of maximizing the rate given a target outage probability. In contrast, the work in the present paper considers this problem for a multiple user, multiple cell system.

4 4 The focus of the present paper is on the systems that use interference averaging in which the average interference on each subchannel on any given link is made equal. One way of achieving interference averaging in OFDM systems is to implement a Latin square design based fast frequency hopping [11, Sect ]. In such fast frequency hopping systems, each fast hopping pattern across the subcarriers represents a virtual subchannel. Virtual subchannels within each cell are orthogonal. Furthermore, during each hopping cycle, any two adjacent cells only use the same subcarrier in the same symbol period exactly once. III. HIGHER LAYER RESOURCE ALLOCATION As part of our two layer approach to solve the resource allocation problem in a frequency selective environment, this section considers the higher layer resource allocation problem. The specific problem associated with the higher layer is to allocate resources to users, to minimize the aggregate transmit power at the base stations while meeting the target data rates of all users. The resources are the subchannels and transmit powers across subchannels. A subchannel here corresponds to a virtual subchannel in a system that employs frequency hopping such as FLASH-OFDM [11], or a physical subcarrier in a system without frequency hopping. In the higher layer problem, we do not address the frequency selective nature of fading; instead, we address this issue at the lower layer in Section IV. A. Model Consider a cellular network which consists of a set of N base stations, denoted by N = {1, 2,..., N}, with each base station n N having a set C n of users. Assume first that the number of subchannels is large compared to the number of users per cell. This allows us to model the apportionment of subchannels in cell n by a continuous weight vector w n R Ln + where L n = C n. This constraint will be relaxed in Section III-E where discretization of the weights is discussed. Each base station is assumed to have the same total number of available subcarriers (and consequently the same number of subchannels). A base station n allocates a proportion w n,m of subchannels to its user m. Thus, w n,m 1. To achieve maximum spectral efficiency, it will be used with

5 5 equality: w n,m = 1. Let w denote the N-tuple of such weight vectors for the network. Normalized bandwidths are used; thus, the weights are both the proportions of bandwidth and the amounts of bandwidth allocated to users. Denote the power density that a base station n allocates to user m C n by ρ n,m. Let ρ n = (ρ n,m ) m Cn be the vector of power densities in cell n. Denote by ρ the N-tuple of such vectors for the network. The power allocated to user m is given by p n,m = w n,m ρ n,m as bandwidths are normalized. Let p n = (p n,m ) m Cn be the vector of power allocation in cell n and p = (p n ) n N be the N-tuple of such vectors for the network. The total transmit power at base station n is q n = p n,m. Let q = (q n ) n N be a vector of total powers for the network. If base stations use a flat power spectrum across all subcarriers, then, q n = ρ n,m, m C n. See Fig. 1 for an illustration. Consider the systems that use interference averaging. Note that interference averaging in a flat fading environment can be achieved either by fast frequency hopping [11], or by using a flat transmit power spectrum at each base station. Denote the receiver noise at mobile m C n by σ 2 m > 0. The path gain from a base station k to user m is Γ k,m. Then, the signal to interference and noise ratio (SIR) at m C n is γ n,m (ρ n,m, q) = Γ n,m ρ n,m σ 2 m + k N,k n Γ k,mq k. (1) Note that since the bandwidth is normalized to 1, σ 2 m and q n are simultaneously powers and average spectral densities. The capacity of a link is determined by both the SIR and the number of subchannels available. For concreteness, we use the Shannon formula for the bit rate: given that the total spectrum allocated to the system is W Hz, and the SIR is γ, the maximum bit rate achieved is W log 2 (1 + γ) bit/sec. The associated spectral efficiency is log 2 (1 + γ) bit/sec/hz. If a user m C n with a bandwidth allocation of W n,m Hz has a rate target of R tar n,m bit/sec, then the rate constraint is Dividing (2) through by W, we get the normalized rate constraint W n,m log 2 (1 + γ) R tar n,m. (2) w n,m log 2 (1 + γ) c tar n,m, (3)

6 6 where w n,m = W n,m / W and c tar n,m = R tar n,m / W (in bit/sec/hz) is the normalized rate requirement. B. Problem Formulation We formulate the higher layer resource allocation problem as a power minimization problem, subject to all users achieving their target normalized rates such that for all n, min ρ,w n N w n,m ρ n,m (4a) w n,m log (1 + γ n,m (ρ n, q)) c tar n,m, m C n, (4b) w n,m = 1, (4c) w n,m > 0, m C n, (4d) ρ n,m = ρ n > 0, m C n, (4e) where γ n,m (ρ n,m, q) is the SIR at user m as defined by (1). The constraint (4e) is the flat power spectrum condition for each base station. As such, ρ n is both the power spectral density (in watts/normalized bandwidth) and the total power (in watts) used by the base station. Thus, the number of degrees of freedom available for base station n is L n : L n 1 weights and a single transmit power level. The above optimization does not necessarily have a feasible solution. In this paper, we assume that the rate targets are chosen such that there is a feasible solution. The problem of ensuring that this is the case is the call admission control problem, and is beyond the scope of the paper. Given feasible rate targets, we provide the resource allocation algorithm to achieve the rate targets. The problem (4) is studied in [12] and there it is shown that if there is a solution, it is unique, and an algorithm is proposed that finds the unique solution when it exists. We review this algorithm in Section III-C, and we will use it to provide the higher-layer resource allocation in this paper. The constraint of a flat power spectrum needs further justification. The justification will be provided in Section IV when we consider frequency selective fading, where this constraint allows an analytical approach to the lower-layer resource allocation problem. In Section III-D we relax the flat power spectrum constraint and allow the power spectrum to vary across each link in the cell.

7 7 C. Resource allocation algorithm: Uniform In this section, we propose a distributed resource allocation algorithm (which we refer to as Uniform ) that provably converges to the optimal solution of (4) when it exists [12]. Initialization: Start with any initial transmit power vector q (0) > 0. For k = 0, 1, 2,... do: Iteration k: Compute a pseudo-weight ŵ n,m for each user m given the power vector q (k) (using (4b)): ŵ n,m = c tar n,m log(1 + γ n,m (q (k) n, q (k) )), m C n. Define ˆσ n = ŵ n,m. Note that the vector ŵ n computed above is infeasible if ˆσ n > 1. Compute a feasible weight vector w (k) n by normalizing ŵ n : w (k) n,m = ŵn,m ˆσ n, m C n. Use the newly computed w (k) n to compute the target transmit power ˆρ n,m for each user m C n (using (4b)): ˆρ n,m = K n,m (q (k) ) [ exp ( ) ] c tar n,m 1, m C w n,m (k) n. Compute the transmit power to use for the next iteration as: q (k+1) n = min ˆρ n,m, if ˆσ n > 1 max ˆρ n,m, otherwise. Each iteration of this algorithm can be considered as a mapping T from q (k) to q (k+1). Although T does not satisfy the monotonicity condition required in Yates framework [13], the convergence of the algorithm can be proved by examining the sequence of generated power vectors. For any sequence (q (k) ) k=0 generated by the algorithm, a monotonically non-increasing upper bounding sequence, and a monotonically non-decreasing lower bounding sequence, can be constructed with the property that both bounding sequences provably converge to the minimal solution. This implies that the sequence (q (k) ) k=0 also converges to the minimal solution. See [12] for the details of this argument.

8 8 Uniform is the result of running the above algorithm to convergence. This is called Scheme 3 in [14]. In Section IV, we will use the allocation provided by Uniform (but with the rate targets adjusted using a rate margin) as input to the lower layer subchannel allocation algorithm (LLSA) which takes account of the fading statistics to solve the lower-layer resource allocation problem. We refer to the combined approach as Layered. D. Alternative resource allocation schemes To provide benchmarks against which to compare the performance of Uniform, we provide two alternative power and bandwidth allocation schemes. Static Bandwidth Allocation: A natural approach to resource allocation is to first fix the bandwidth allocation based on the rate targets, and then use power control to try to achieve the rate targets. We refer to this approach as Static. In Static, the weight vectors w n are chosen a priori, proportional to the normalized rate targets of the users, i.e., w n,m = c tar n,m m C n c tar n,m, m C n. (5) The resource allocation problem then is to choose the appropriate powers for users with the objective of power minimization, subject to meeting the normalized rate requirements of the mobiles. Mathematically, the allocation problem is to solve the optimization problem subject to (4b) and also min ρ n N w n,m ρ n,m (6) ρ n,m > 0, m C n. (7) Under Static, the number of degrees of freedom available to base station n is again L n : there are L n powers to allocate to the mobiles in the cell. The solution to the static subchannel allocation problem can be found using the Yates framework [13], as shown in [14]. Note that this scheme requires some form of signal hopping over subcarriers, similar to FLASH- OFDM, in order to achieve the effect of interference averaging. This is not required by Uniform, at least in the flat fading scenario of the present section.

9 9 Dynamic Bandwidth and Power Spectral Densities: Here, we propose a scheme that is more ambitious than either Static or Uniform, and in effect tries to combine the flexibility of both. We refer to this approach as Dynamic. Dynamic solves the optimization problem: (4a) subject to (4b), (4c), (4d) but with (4e) replaced by (7). With Dynamic, the number of degrees of freedom available to base station n is increased to 2L n 1: L n power densities and L n 1 weights. This problem can be solved using an iterative algorithm that makes use of the Yates framework [13], as in the case of Static. This iterative approach is motivated by the fact that, in order to minimize the total transmit powers of the network, the transmit power in each cell must be minimized. This problem can be decomposed into a set of coupled single cell subproblems, each of which is convex. The subproblem of minimizing the transmit power in a cell for a given interference level forms the basis of the iterative algorithm. Refer to [14] for the details of the development of the algorithm. E. Discrete subchannel allocation So far, the allocation of subchannels to users has been treated as continuous. The following rounding procedure is then applied to obtain discrete subchannel allocations to users. The rounding procedure is the same for all three schemes. The procedure starts with the list of weights to be discretized. The discretization is done one at a time, starting with the smallest weight. The smallest weight in the list is rounded up to the nearest discrete value and is taken off the list. The weights that are still in the list are then decreased by an equal factor so that the weights sum to unity again. This procedure is repeated until the list is empty. The rounding procedure described above can exactly be applied to Static before the power control algorithm is run, due to the static nature of the bandwidth allocation in Static. For Dynamic and Uniform, the discretization of the weights can only be performed after the power control algorithm is run. For Dynamic, we update the power levels after the discrete weight vectors have been found, using an algorithm similar to that of Static. We do not attempt to update the power levels for Uniform in this way because of its flat power spectrum requirement. When Uniform is used with

10 10 the LLSA algorithm of Section IV, the weights obtained during the higher layer allocation become superfluous; the LLSA algorithm generates a discrete subchannel allocation. IV. RESOURCE ALLOCATION UNDER MULTIPATH FREQUENCY SELECTIVE FADING In the preceding section, we considered the resource allocation problem for an OFDM cellular network in a static environment in which the channel gains do not change. In a frequency selective fading environment, both signal and interference experience fluctuations, resulting in variations in the achieved rate at the receiver. Our basic assumption is that frequency selective fading is too fast to track at the base station, and that the base station only has statistical knowledge of the fading. In a fading environment in which the base stations do not have perfect channel knowledge, a natural measure of performance is the outage probability. In this paper, we assume that there is coding over the subchannels used by any particular user, and a user will be in outage if the total mutual information between sent and received signals summed over the allocated subchannels falls short of the threshold needed to support the user s target data rate. The performance of a resource allocation algorithm can then be quantified by the probability of outage for each of the users. Any one of the three deterministic schemes developed in Section III could be used to obtain a resource allocation in the frequency selective fading environment by simply replacing the random gains with their averages. However, when they are applied directly by using the average values of the random gains, none of the three schemes has good performance with respect to the outage probability metric. This should come as no surprise, as it has long been recognized that fade margins are required to account for statistical fluctuations in communication systems. The user outage probabilities can be improved by employing a rate margin. An appropriate margin can be added to the users normalized rate targets during the running of the deterministic algorithm. This in effect corresponds to targeting higher normalized rates for the users. Ideally, appropriate rate margins should be applied to each user so that the variation across the user outage probabilities is minimized. Unfortunately, it does not seem at all straight forward to predetermine appropriate rate margins for each user individually in our case; at least, it is beyond the scope of this paper.

11 11 Therefore, we provide a two timescale, layered approach to the resource allocation problem. The first (higher) layer deals with the allocation of transmit powers to users by considering the average gains of the links. The lower layer addresses the issue of minimizing the variation of outage probability values across the users. Uniform is used at the higher layer to allocate transmit powers to cell sites. It provides the appropriate structure for the development of the lower layer resource allocation algorithm. During the lower layer allocation, the total transmit power in each cell is kept fixed to the value obtained by the higher layer allocation. This enables the lower layer algorithm to operate independently in each cell, without knock-on effects between cells. Note that, in a frequency selective fading environment, employing fast frequency hopping together with Uniform simplifies the design problem; by doing so, we only need to model the number of subchannels for each user, as in the case under flat fading. Uniform is implemented on a slow timescale to track large-scale changes in the network, such as changes in cell loadings (both from call arrivals and departures, and handover between cells), and large-scale mobility of users within a cell. On a faster timescale, subchannels are allocated between users within the cell using the proposed subchannel allocation algorithm. The proposed algorithm aims to obtain a subchannel allocation that minimizes the maximum outage probability among the users, given the total transmit power in each cell. The specific statistics used in the optimization are the mean and the variance of the achieved normalized capacity of a subchannel. Each receiver will measure these statistics over an integer number of hopping cycles and feed them back to its base station. Note here that since the number of subchannels is a discrete quantity it may not be possible to obtain an allocation that exactly equalizes the outage probabilities among all users. A. Subchannel Allocation - Problem Formulation We consider a simplified version of the subchannel allocation problem where the fading on the subchannels within a link are independent and identically distributed (i.i.d.). This may represent the case in which no user gets allocated a large proportion of the subchannels and hence the average correlation between subchannels on a given link will be very low, and can be neglected.

12 12 Let N c be the number of subcarriers in the system, and consider a cell n. Let η m be the number of subchannels allocated to user m C n ; since the allocation of subchannels in cell n does not depend on the subchannel allocations in other cells (as the powers are fixed), we drop the label n from the notation at this point. Let η = (η m ) m Cn. The instantaneous gain of a subchannel i consists of two components: Γ n,m which is the average gain of link [n, m], and F (i) n,m which models fading on subchannel i. The instantaneous signal to interference and noise ratio (SIR) on subchannel i is γ (i) n,m = Γ n,m q n F (i) n,m σ 2 m + k n Γ k,mq k F (i) k,m The corresponding instantaneous normalized rate of subchannel i is. (8) c (i) m = 1 N c log(1 + γ (i) n,m). (9) The achieved normalized rate of user m is the sum of the achieved normalized rates of individual subchannels allocated to that user: η m i=1 c(i) m. The outage probability is ( ηm ) Pm out (η m ) = P c (i) m < c tar m. (10) The problem of minimizing the maximum outage probability is i=1 min η max P out m (η m ) (11a) such that η m = N c. (11b) Solving (11) exactly will require the knowledge of the distribution of c (i) m s. However, a good approximation can be made using only the mean and variance of the random sum. Let Z m = η m i=1 c(i) m, E[c (i) m ] = µ m and variance of c m (i) be β 2 m. Define, Ẑ m (η m ) = Z m η m µ m ηm β m. (12) The outage probability of user m is given by ˆP out m (η m ) = P (Ẑm(η m ) < B m (η m )) (13)

13 13 where B m (η m ) = ctar m η m µ m. (14) ηm β m By the central limit theorem, the Ẑm(η m ) are approximately identically distributed for sufficiently large η m. Note that this approximation concerns the similarity in the distribution of the Ẑm to each other, and not their similarity to the Gaussian. From the right hand side of (13), minimizing the maximum of ˆP out m (η m ) is then equivalent to minimizing the maximum of B m s among the users. Thus, the optimization problem to solve is such that min η max B m (η m ) (15a) η m = N c, (15b) η m Z +, m C n. (15c) B. Lower Layer Subchannel Allocation (LLSA) Algorithm The problem (15) is a nonlinear integer programming problem, which is combinatorial in nature. There may be multiple allocations solving it (existence of a solution is guaranteed as long as N c L n, with L n being the number of users in the cell n). We now derive an O(L n ) algorithm that solves (15). We will refer to the following development as Algorithm (LLSA). It is easier to first solve the continuous relaxation of (15), by dropping the integrality constraints on the number of subchannels allocated to each user. The continuous problem is such that min x max B m (x m ) (16a) x m = N c, (16b) x m R +, m C n. (16c) where the x m are not required to be integers and x = (x m ) m Cn.

14 14 Lemma 1: The problem (16) has a unique solution x, and B m (x m) = B for all m C n. Proof: The inverse of B m ( ) is x m (B) = ( (Bβm ) 2 + 4µ m c tar m 2µ m Bβ m ) 2 (17) where the + has been taken in the quadratic implied by (14) since x m > 0. Since x m ( ) is strictly decreasing, it follows that x m (B) is also. Thus x m (B) = N c has a unique solution B = B since x m ( ) = and x m ( ) = 0. Thus the unique solution to (16) is x m = x m ( B ) for all m C n, due to the monotonicity of B m ( ). We note that B is the minimum of the relaxed problem (16). Problem (16) can be solved by first determining the value of B and then using it to find x. The value of B can be determined by a bisection search. Initial bounds can be constructed by selecting an x > 0 which satisfies x m = N c, and then setting B u = max x m (B u ). B m (x m ) and B l = min B m (x m ), giving x m (B l ) N c Let B be the minimum of the original problem (15). Then, B B. Now define a vector η = ( η m ) m Cn with η m = x m ( B ), m C n. (18) Since η m x m ( B ) and B m (.) is monotonically decreasing, B m ( η m ) B B, m C n. Furthermore, η m N c. If η m =N c then the problem (16) has an integral solution, which solves (15). Turning our attention to the typical case that η m N c + 1, note that B m ( η m ) B, m C n. Thus, we can apply the following lemma to reduce the excess in the number of allocated subchannels. Lemma 2: For any η = (η m ) m Cn, if B m (η m ) B, m C n, and η m N c + 1, then, min B m (η m 1) B. (19)

15 15 Proof: Let η = (η m) m Cn be a subchannel allocation that solves (15) (not necessarily unique). Since η m N c + 1 and ηm = N c, there exists a user m C n such that (η m 1) ηm. Since B m(.) is monotonically decreasing, B m (η m 1) B m (ηm ) B. An integral solution of (15) can be constructed by successively applying Lemma 2, starting with the allocations in η, and terminating after at most L n steps in such a solution. We start with η (0) = η and note that this allocation involves r = η m N c excess subchannels, which need to be removed. We do this iteratively. Suppose η (k) has r k > 0 excess subchannels, and satisfies B m (η (k) m ) B, m C n. Then Lemma 2 applies. In particular, removing a subchannel from user m that satisfies m = arg min B m (η (k) m 1) results in a new subchannel allocation η (k+1) that has r (k + 1) excess subchannels, and satisfies B m (η m (k+1) ) B for all m C n. By induction, we obtain after r steps that there are no excess subchannels, so we must be at a solution to the problem (15). We have therefore provided an algorithm, namely the Lower Layer Subchannel Allocation (LLSA) algorithm, that finds an integral solution to the subchannel allocation problem. We first solve the relaxed problem, using the bisection method. We then take a finite number of steps (at most equal to the number of users in the cell) to find an integral solution. Note that the algorithm is not equivalent to substracting one subchannel each from the r users with the smallest B m ( η m 1), as there are cases where more than one subchannel must be removed from the same user as in the following example. Consider a cell with L n = 3 users and N c = 7 subchannels. The B m ( ) values for the cell are given in Table I. Suppose that the solution to the relaxed problem is x = [4.2, 1.6, 1.2], giving an initial allocation of [5, 2, 2]. The corresponding values of B m are [0.5, 2.2, 1.9]. There are r = 2 excess subchannels to be removed. Applying the LLSA algorithm will result in 2 subchannels to be removed from user 1 resulting in an alocation η = [3, 2, 2], which is optimal. From Table I, the corresponding values of B m are [1.8, 2.2, 1.9], with a maximum value of 2.2.

16 16 Of our three deterministic schemes, it is only practical to apply the LLSA algorithm to Uniform, because the central limit theorem approximation assumes that all subchannels have the same mean and variance, independent of the subchannel allocation, and this mean and variance needs to be measured. With Dynamic or Static, one could average the mean and variance across subchannels, but the values obtained would still be a function of the subchannel allocation itself. Furthermore, in Uniform, the total power allocated to the base station does not change as a function of the subchannel allocation, whereas it does change for Dynamic and Static. C. Numerical Results This section numerically evaluates the performance of the proposed two layer approach. Consider a cellular network that spans an area of 3km 3km. There are 9 equal-sized square cells of size 1km 1km. The base stations are located at the center of their respective cells. There are 90 users, uniformly randomly distributed in the network. Each user communicates only to the base station of the cell it is located in. The normalized rate target for each user is chosen uniformly at random from a set {c, 2c, 3c, 4c}, where c is a scaling factor. The normalized rates can be varied by tuning the scaling factor c. Log-distance path loss [15] with a path loss exponent of 3 is used (with the reference distance set to 500 m). The log-normal shadowing has a mean of 0 db and a standard deviation of 8 db for all distances. The noise spectral density is W/Hz at each receiver. A network realization in this context refers to a random realization of user locations and shadowing. The system uses explicit interference averaging. Fast frequency hopping patterns based on Latin squares were implemented for this purpose. The subchannel allocation to each user is discrete. There are N c = 113 subcarriers in the system. As the FFT implementation requires the number of subcarriers to be a power of 2, this can correspond to a system with 128 subcarriers, with the remainder to be used as pilot and null subcarriers. Within each link, the subcarrier gains are independent and Rayleigh distributed about the flat fading gain. We consider here the fast fading case in which the subcarrier gains change on a faster time scale than that of the hopping cycle. We refer to the scheme corresponding to our two layer approach as Layered. For Layered,

17 17 the cell level transmit powers are determined using Uniform. The LLSA algorithm is then applied to find the subchannel allocation. We compare the outage performance of Layered with that of Dynamic, Static and Uniform. Dynamic, Static and Uniform use the resource allocations determined by the deterministic algorithms with fade margins, including rounding to the discrete subchannels, as described in Section III. These schemes provide a point of comparison, to test the effectiveness of the LLSA algorithm applied to Uniform. Monte Carlo simulations were used for calculating the achieved outage probabilities of the users. We first study the average power and outage performances of the schemes with varying normalized rate targets. For each network realization, the normalized rate targets can be varied by tuning the scaling factor c. As the capacity limit for different network realizations are achieved at different values of c, the average total symbol energy vs. average total normalized rate curve of Fig. 2 is obtained as follows. For each network realization, the resource allocation was performed for the schemes for a range of values of c. Then, for a fixed set of total symbol energies, corresponding total normalized rates are computed by linear interpolation. The average total normalized rate for a given total symbol energy is the average of such computed total normalized rates over many network realizations. Fig. 2 plots the average total symbol energy versus average total normalized rate, averaged over 50 network realizations. Fixed rate margin values of 25%, 50% and 100% were applied. From Fig. 2, it is clear that Dynamic uses the least amount of energy for a given total normalized rate. This is to be expected as Dynamic solves the optimal allocation for the flat fading scenario. The loss in performance in terms of capacity for a given total symbol energy is relatively small for both Static and Uniform. Uniform marginally outperforms Static. The corresponding curve of the mean of the maximum outage probability versus the mean total normalized rate is plotted in Fig. 3. From Fig. 3, we observe that the outage performance of the network in fact improves with the user rate targets for Dynamic and Uniform, while the maximum outage probability does not vary significantly for Static and Layered. Furthermore, increasing the rate margins indeed benefits all schemes. Layered provides a significant reduction in outage

18 18 probability, particularly at outage probabilities of practical interest, e.g., below 10%. We now investigate further the relationship between the rate margins and the outage performance of the users. We use a fixed value of c = for this purpose. With a total bandwidth of 10 MHz, this corresponds to bit rates of 360, 720, 1080, and 1440 kbits/sec. Each user randomly picks one of these bit rates, as above. By varying the rate margin we can vary the total symbol energy (in watts/hz). The values of rate margins were varied in the range of 0% 50% (factors from 1 to 1.5). A given rate margin results in a different total symbol energy for each scheme. For each rate margin, computation of outages over 50 network realizations were performed, and the mean value of the maximum user s outage probability is plotted against the mean total symbol energy in Fig. 4. Notice that the outage performance of all schemes improve with the rate margin. The performances of Dynamic, Static and Uniform are similar. With smaller rate margins, Uniform has the worst performance as a user with high average path gain may be allocated a very small number of subchannels in this scheme, and this does not provide as much frequency diversity as achieved by, say, Static. Dynamic can overcome this by allocating more power spectral density to the links that do not receive many subchannels. At high rate margins, Static is the worst, and Uniform achieves very similar performance to Dynamic. All three schemes suffer diminishing returns as the rate margin is increased. The use of LLSA does not seem to greatly improve the performance of Uniform at high outage probabilities. However, the performance of Layered improves greatly as the rate margin is increased, showing the usefulness of our two layer approach when the outage probabilities are below, say, 10%. Since outage probability targets are typically below this level, our two layer resource allocation scheme is a promising approach. Fig. 5 plots the mean achieved outage probability of the users as a function of total symbol energies. In this case, the penalty incurred by Uniform at high outage probabilities is not observed, as the under-allocation of subchannels to some users is balanced by the over-allocation to others. Furthermore, Layered has better mean outages, even though the objective was to minimize the

19 19 maximum outage. This could be explained as follows. The subchannels are typically removed from users with large initial allocations ( donors ) and given to users with small initial allocations ( recipients ). The fractional change in the number of subchannels is larger for the recipients, and so their expected decrease in outage will be larger than the increase in outage of the donors. We now compare the distribution of user outage probabilities for all schemes. For a particular network realization, Fig. 6 plots the outage probabilities of users grouped by cells, with the probability values sorted in descending order in each cell. The rate margins of 28.5% ( Dynamic ), 21.5% ( Static ) and 25% ( Uniform ) were chosen to make the total symbol energy equal to W/Hz for all schemes. From Fig. 6, it is clear that the LLSA algorithm reduces both the outage variations and total outage among the same cell users compared to the other three schemes. V. CONCLUSIONS This paper has considered the resource allocation problem in a multiple user, multiple cell OFDM cellular network under frequency selective fading. A two layer approach was proposed to tackle this problem. In the higher layer problem, the power allocations are obtained using the average gains of the links. A uniform power spectral density scheme is chosen as the method of choice at the higher layer which facilitates the development of the lower layer algorithm. The second layer of the approach allocates numbers of subchannels to users. This allocation is performed by the LLSA algorithm within each cell in order to minimize the maximum outage probability. This balancing of outage probabilities is done independently in each cell using the locally measured channel statistics, independent of other cells. The simulation results show that the LLSA algorithm works very well in balancing the outage probabilities among the users. A key contribution of this paper is a method of resource allocation that avoids a combinatorial explosion as the number of users and subcarriers grows large. In particular, the higher layer problem is not combinatorial (it is a continuous relaxation of a combinatorial problem) and it converges exponentially fast, as is typical for these sorts of power control algorithms. The lower layer subchannel allocation problem is combinatorial, but is approximated using the central limit theorem. The use of

20 20 interference averaging is crucial at this step. This paper uses the uniform power spectral density allocation scheme ( Uniform ) at the higher layer and shows that Uniform enables the development of a simple per-cell outage equalization at the lower layer which significantly improves the user outage performances. It may also be possible to devise lower layer algorithms when Dynamic or Static is employed at the higher layer. REFERENCES [1] ITU-T Recommendation G.992.1, Asymmetrical digital subscriber line (ADSL) transceivers, Jul. 1999, also Annex H (10/2000) and Corrigentum 1 (11/2001). [2] ITU-T Recommendation G.993.1, Very high-speed digital subscriber line foundation, Nov [3] IEEE Std g, Wireless LAN medium access control (MAC) and physical layer (PHY) specifications, amendment 4: Further higher data rate extension in the 2.4 GHz band, Jul [4] D. J. C. MacKay, Elements of Information Theory, 1st ed. John Wiley & Sons, [5] W. Yu, W. Rhee, S. Boyd, and J. M. Cioffi, Iterative water-filling for Gaussian vector multiple-access channels, IEEE Trans. Info. Theory, vol. 50, no. 1, pp , Jan [6] D. Kivanc, G. Li, and H. Liu, Computationally efficient bandwidth allocation and power control for OFDMA, IEEE Trans. Wireless Commun., vol. 2, no. 6, pp , Nov [7] Z. Shen, J. G. Andrews, and B. L. Evans, Adaptive resource allocation in multiuser OFDM systems with proportional rate constraints, IEEE Trans. Wireless Commun., vol. 4, no. 6, pp , Nov [8] G. Kulkarni, S. Adlakha, and M. Srivastava, Subcarrier allocation and bit loading algorithms for OFDMA-based wireless networks, IEEE Trans. Mobile. Comput., vol. 4, no. 6, pp , Nov./Dec [9] N. Damji and T. Le-Ngoc, Adaptive BCPM downlink resource allocation strategies for multiuser OFDM in cellular systems, in Proc. IEEE International Conference on Communication and Broadband Networking (ICBN), vol. 1, Oct. 2005, pp [10] Y. Yao and G. B. Giannakis, Rate-maximizing power allocation in OFDM based on partial channel knowledge, IEEE Trans. Wireless Commun., vol. 4, no. 3, pp , May [11] D. Tse and P. Viswanath, Fundamentals of Wireless Communication, 1st ed. Cambridge University Press, [12] T. Thanabalasingham, S. V. Hanly, and L. L. H. Andrew, Joint allocation of transmit power levels and degrees of freedom to links in a wireless network, in Proc. Australian Commun. Theory Workshop, Feb. 2005, available at IEEE Xplore. [13] R. D. Yates, A framework for uplink power control in cellular radio systems, IEEE J. Select. Areas Commun., vol. 13, no. 7, pp , Sep [14] T. Thanabalasingham, S. V. Hanly, L. L. H. Andrew, and J. Papandriopoulos, Joint allocation of subcarriers and transmit powers in a multiuser OFDM cellular network, in Proc. IEEE Int. Conf. Commun. (ICC), Jun [15] T. S. Rappaport, Wireless Communications, 2nd ed. Prentice Hall, 2002.

21 21 Fig. 1. The power allocation p n,m to user m C n depends on power density ρ n,m and the proportion w n,m of bandwidth (subchannels) allocated to that user, p n,m = w n,mρ n,m. Pictorially, it is the area of the rectangle corresponding to user m. The total power q n at base station n is the total area under the curve. With a flat power spectrum, q n = ρ n,m, m C n since P w n,m = Mean total symbol energy (W/Hz) % 50% 25% "Dynamic" "Static" "Uniform" Mean total normalized rate (bits/sec/hz) Fig. 2. Mean total symbol energy vs. mean total normalized rate (averaged over 50 network realizations). Fixed rate margins of 25%, 50% and 100% were applied for all schemes. Note that Layered uses Uniform for its power allocation, so its performance is given by Uniform above.

22 Mean of the maximum outage probability Dynamic" "Static" "Uniform" "Layered" Mean total normalized rate (bits/sec/hz) (a) 25% 10 0 Mean of the maximum outage probability Dynamic" "Static" "Uniform" "Layered" Mean total normalized rate (bits/sec/hz) (b) 50% 10 0 Mean of the maximum outage probability Dynamic" "Static" "Uniform" "Layered" Mean total normalized rate (bits/sec/hz) (c) 100% Fig. 3. Mean of the maximum outage probability vs. mean total normalized rate (averaged over 50 network realizations). Fixed rate margins of 25%, 50% and 100% were applied for all schemes.

23 "Dynamic" "Static" "Uniform" "Layered" Maximum outage probability Total symbol energy (W/Hz) x 10 8 Fig. 4. Mean over 50 network realizations of the maximum outage probability (over users). The total symbol energy increases with the rate margin applied "Dynamic" "Static" "Uniform" "Layered" Mean outage probability Total symbol energy (W/Hz) x 10 8 Fig. 5. Mean over 50 network realizations of the mean outage probability (over users). The total symbol energy increases with the rate margin applied.

24 24 η m m TABLE I VALUES OF B m (η m ) FOR A CELL WITH 3 USERS AND 7 SUBCHANNELS. FOR AN INITIAL SUBCHANNEL ALLOCATION OF [5, 2, 2], THE CORRESPONDING VALUES OF B m ARE 0.5, 2.2 AND 1.9. THE OPTIMAL ALLOCATION IS [3, 2, 2] WITH CORRESPONDING B m VALUES OF 1.8, 2.2 AND 1.9. Outage probability "Dynamic" "Static" "Uniform" "Layered" User index Fig. 6. Outage performance of the users for a particular network realization. The rate margin for each scheme was selected to make all schemes use a same total symbol energy of W/Hz. The outage probabilities of same cell users are sorted in descending order.